Proof of the Law of Extensions

We want to show, for an arbitrarily chosen concept P and
an arbitrarily chosen object c, that c ∈
εP ≡ Pc.

(→) Assume c ∈ εP (to show
Pc). Then, by the definition of ∈, it follows
that

∃H(εP = εH &
Hc)

Suppose that Q is such a property. Then, we
know

εP = εQ & Qc

But, by Basic Law V, the first conjunct implies
∀x(Px ≡ Qx). So from the fact
that Qc, it follows that Pc.

(←) Assume Pc (to show c ∈
εP). Then, by the Existence of Extensions
principle, P has an extension, namely, εP.
So by the laws of identity, we know εP =
εP. We may conjoin this with our assumption to
conclude